CAIE IGCSE Mathematics Extended Trigonometry Questions

Practise solving linked two-dimensional triangle problems with Pythagoras and trigonometry, including shortest distances and derived sides.

Exam points

select Pythagoras or a sine, cosine or tangent ratio from the known sides and angles
calculate an intermediate side or angle before applying it in a second linked triangle
recognise a shortest distance as the perpendicular from a point to the target line

Covered syllabus points

E6.2.1 Know and use the sine, cosine and tangent ratios for acute angles in E6.2.2 Solve problems in two dimensions using Pythagoras’ theorem and E6.2.3 Know that the perpendicular distance from a point to a line is the E6.2.4 Carry out calculations involving angles of elevation and depression

[Maximum number: 7]

(a)

NOT TO SCALE

The diagram shows a field, ABCD with B north of A.
BD is a path across the field.
$A B=85 \mathrm{~m}, A D=72 \mathrm{~m}, B D=129 \mathrm{~m}$ , angle $B D C=39^{\circ}$ and angle $B C D=60^{\circ}$ .

[ 7 ]

(i)

Show that angle $A B D=31.6^{\circ}$ , correct to 1 decimal place.

[ 4 ]

$[\cos=]\frac{85^2+129^2-72^2}{2\times85\times129}$
M2
M1 for $72^2=85^2+129^2-2\times85\times129\cos ABD$ $31.58\ldots$
A2
A1 for 0.851 to 0.852 or $\frac{9341}{10965}$ or equivalent fraction

(ii)

Find the shortest distance from A to BD.

[ 3 ]

44.5 or 44.51 to 44.54
M2 for implicit correct method, e.g. $\frac{d}{85}=\sin31.6^\circ$ oe
or M1 for recognition that the line from A is perpendicular to BD

[Maximum number: 5]

NOT TO SCALE

The diagram shows triangle ABC on horizontal ground.
$A C=15 \mathrm{~m}, B C=8 \mathrm{~m}$ and $A B=20 \mathrm{~m}$ .
BP and CQ are vertical poles of different heights.
$B P=3 \mathrm{~m}$ and $C Q=4 \mathrm{~m}$ .
AQ and PQ are straight wires.

(a)

Calculate the length of AQ.

[ 2 ]

15.5 or 15.52 to 15.53
M1 for $15^2+4^2$ oe

(b)

Calculate the angle of elevation of Q from P.

[ 3 ]

7.1 or 7.13 or 7.125 to 7.126
M2 for $\tan P=\frac{4-3}{8}$ oe or M1 for vertical line =4-3

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Covered syllabus points

E6.2 Right-angled triangles question 1

Question (a)

Question (i)

Question (ii)

E6.2 Right-angled triangles question 2

Question (a)

Question (b)